Strokes77:
Your Black Taurus with the bugs stuck on it are a very common "paint problem" that owners face on their vehicles. Unfortunately, automatic car washes (IE, touchless car washes) that most vehicle owners use at gas stations or quik-car washes DO NOT remove the bug residue from these bug impacts. Here in the upper Midwest, grasshoppers and butterflies are probably the two worst insects that cause the most damage as they really "stick" to the front facias (bumber and grills) and can REALLY etch the paint over time if left there, especially in the hot summer sun. This is one of the reasons most Autopians do a hand-wash of their vehicles on a regular basis.
I see this "etching" all the time, and quite frankly, if you are trying to remove the worst of it by hand, you may get some of it out, but it will still look "bad" by most anyone's measure of viewing the paint.
As stated in previous posts, claying is probably THE most acceptable method to begin the contaminant removal process after washing. One suggestion about washing: DON'T attempt to vigorously SCRUB off the bugs with a bug pad. I've seen "uneducated/ignorant" owners attempt to use Brillo pads, Scotch-Brite pads, or Comet scouring powder to remove baked-on bugs, and yes, it DOES remove them, but it scratches the living day-lights (that's the politically correct term) out the the paint or chrome, and it's impossible to fix short of replacing the chrome or repainting the bumpers.
Once you've removed the bug contaminants from the paint, chances are they have etched the paint (actually, it's the clearcoat) and attempting to polish these out by hand is an almost futile physical job. Unless you are human in great shape, the "elbow grease" required to do so is exhausting, not to mention the patience it requires. It probably won't come out with 30 seconds of rubbing with over-the-counter Meguiar's Scratch-X 2.0. I am not trying to "bust your bubble": it's the reality of the effort.
This is one reason many Autopians have invested in a Porter-Cable Dual-action Random Orbital machine polisher (or "PC" as its ancromyn is noted here on Autopia). Use of a machine and pads and proper polishes will achieve results that hand-polishing could never achieve without the all the exhausting, physically-demanding "elbow grease". This is the next step after using the clay to remove or at least mitigate the etching on your bumper. If you can't or don't want to make an investment in one of these, you might find a fellow Autopian near where you live who can help you out.
Removing bugs from Chrome grills/trim is tricky and I find a more-abrasive chrome polish, like Simi-Chrome or MAAS metal cream, will work with a little elbow grease. On grills or trim, though, these are chromed-PLASTIC that easily scratches, so you will need to "adjust" how much elbow grease is required. On bumpers I use a true chrome cleaner, like Simonize chrome cleaner (I have an old formula thats like liquid sandpaper in a can) as it's chromed metal and more resistant to scatches.
On hard plastic trim, I use a tooth-brush and and a good All-Purpose Cleaner (APC), like Meguiar's Detailing Line D-101, (Yes, this can be purchased from an Auto Parts store). Just DON'T use Simple Green as an APC; it can wreak havoc on aluminum pieces so common on newer cars. Then treat it with a plastic trim protectant. I use Meguiar's Ultimate Protectant, but it's very finicky and tends to streak on some plastics (I think I have an early production batch that did have this problem). Some here use S100 Carnauba paste wax found at Harley-Davidson motorcycle shops (Great wax to have in your dealing supplies collection) or Collinite 845 Insulator Wax (845IW) (another must-have wax for do-it-yourselfer's) on plastic trim. I know that using a wax on plastic is counter-intuitive to what has been thought or experienced in the past, but you can try it and see if it works for you.
Yes, this is more-than-you-really-wanted-to know, but I hope it helps. Keep reading and searching this forum. You'll gain a wealth of experience and understanding, and perhaps you, too, will become an Autopian.